Deriving space-time in four-dimensional Euclidean space with no time and dynamics

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Deriving space-time in four-dimensional Euclidean space with no time and dynamics

About year and half ago I started thread in the ATM forum with proposal of unified field theory: "Spacetime-and-matter-as-emergent-phenomena-unified-field-theory". It was very useful discussion.
Thanks to Reality_Check and other participants.

The discussion shown to me that in my theory distance between start point and results, such as QM etc, it too big. After some thinking of results of the discussion, I decided to write article with foundations of the theory. It contains only foundations, quantum mechanics etc is not considered at all in the article. Later, I presented the article with foundations of the theory on several seminars and one international conference on physics. I cannot say it caused lots of interest, but some discussions happened. After one of seminars, in following email discussion, was found quite significant error in equations, I fixed it later. As of now, I not know any error in the article.
The article, as far as I see, cannot be published in physics journals because it challenge realism. Editors from physics journals responds it should go to philosophy journals, editors from from philosophy journals write it is physics and should go to physics journals.

What is in the article?
The article contains foundation of discussed earlier theory. It not contains any quantum mechanics, it only contains basis of theory.

In the article, I propose following model:
1. Time and dynamic is absent on fundamental level. No any motion, no energy, nothing related to time and dynamic on fundamental level
2. On fundamental level there is Euclidean space, with al least 4 dimensions. (And yes, I know about impossibility to derive hypersurface with Lorentz metric in Euclidean space. There is solution for the theory)
3. All dimensions are equal, there is no preferred direction.
4. Reiterating that was written before – time and dynamic on fundamental level is absent. Completely. No anything like time dimensions etc.
5. There is some field or field on fundamental level. The field(s) are defined at each point of fundamental space and have values belonging to set of real numbers (scalar field). (Scalar fields, described in textbooks for QFT, have different properties than these fields, so statement about insufficient degrees of freedom is not applicable here. But lets put it aside of the discussion) There is no time or dynamics. Thereby, the fields also have no dynamics. It also means full determinism. I will call these fields fundamental ones. I suppose that the fundamental fields are smooth and are described by certain partial differential equations. Each of the fundamental fields is independent of other fundamental fields. This means that there are no other fields in the equations describing any fundamental field.
6. Quite obviously, it is not possible to add observer to the model in traditional way. Observer always requires time for its existence. Absence of time means it is necessary to add something else to add observers. Instead of time dimension, I use space dimension. Details are in article. All space dimensions, as I already write, are equal, no preferred direction. Observer is able to observe changes because I postulate that changes on consecutive 3-d hyperplanes in fundamental space can lead to appearance of observer. [These is hardest of understanding point of the model].
7. Because observer appear as result of changes of field(s) on consecutive 3-d hyperplanes in fundamental space (I reiterate, there is no changes in fundamental space, But state of projections of fundamental field(s) on consecutive hyperplanes can change), observer is not exists objectively. And even more, Universe is not exists objectively. It exists only when there is some observer which observe it. Without observer, spacetime in the model is just mathematical abstraction.
So, I propose subjective idealism in foundation of my theory. Fundamental space with defined on the space field(s) exists objectively. But, because observer cannot exists without time and dynamic, the space and fields exists in quite nontraditional way, without any ability for direct observation. Their presence can be verified only indirectly, based on how well the theory fit to observations.

As one can notice, there is no relativism at the model. There is no aether at the model. There is no motion at the model. There is no gravity at the model.
What I claim as done in the article in scope of the theory:
1. Derived anthropic principle. Yes, derived, not postulated
2. Derived principle of causality
3. Derived equations of special relativity
4. Derived principle of locality
5. Found what is gravity
6. Derived equations of general relativity. And I derived in in such way, that there is clear explanation why gravity part is absent in tensor of energy-mass.

And all above done on model without time, without dynamic, without principle of locality, without gravity.
So, I remove lots of phenomena from list of fundamental ones.

The claims, as it can be seen, are quite big. I am interesting in testing the theory, test are the results correctly derived, are any obvious weaknesses.

One moderational note: this thread stands on its own, only the topic in the OP will be discussed.
No links to the previous thread by Ans are allowed.
Therefore, I also deleted the link at the beginning of the OP.
Have fun discussing.

....1. Derived anthropic principle. Yes, derived, not postulated
2. Derived principle of causality
3. Derived equations of special relativity
4. Derived principle of locality
5. Found what is gravity
6. Derived equations of general relativity. And I derived in in such way, that there is clear explanation why gravity part is absent in tensor of energy-mass.

This is wrong, Ans.
The anthropic principle is not a scientific principle that can be derived. It is a philosophical consideration. We note that the universe allows for life in the current era and that we exist.
As soon as you write "four-dimensional Euclidean space" you cannot derive anything in relativity because it is mathematically impossible for relativity to exist in a Euclidean space. That is especially true for general relativity.
"No time" is extremely wrong because time is what separates event. For example it took time separated the start of my typing rom the end of my typing. SR and GR explicitly have time as a 4th dimension. No time = no time dimension and makes it is impossible to derive anything in them.

The PDF has:
"There are two principal models of the nature at current time." which is wrong. There have been no aether models in mainstream physics for 115 years.
"Let's assume that at the fundamental level time does not exist at all." is the "no time" error. We cannot assume that because that is not this universe. We see time at all fundamental levels. We see dynamics at all fundamental levels. You acknowledge this indirectly by writing "change" that needs time.
Non sequitur: An unclear "consciousness is an epiphenomenon caused by the change of physical fields on hypersurfaces" postulate does not lead to the anthropic principle. At best it is trivial logic "consciousness exists thus consciousness exists".
"Principle of Causality" is unclear assertions that are not about causality in physics.
"Time Vector Direction" is word salad, not physics.
"Derivation of Hypersurfaces and Observer" is word salad, not physics.
"Symmetry to the Translations of the Emergent Time and Space" is word salad, not physics.
"Observable Physical Fields" is word salad, not physics.
"Inertial Frames of Reference" starts with a over-simplified definition that ignores your "no time" requirement. If there is no time then there is no speed and no inertial frames of reference! "The question now arises of how to move from one inertial frame of reference into another." followed by nonsense. The way to move from on frame to another is to change speed. "The first consequence, relativity of simultaneity" is not the relativity of simultaneity. Word salad about observed time dilation.
"Energy" is word salad, not physics. Energy has a definition in physics.
"Hyperplane Velocity and Angle of Rotation" ignores your "no time" requirement. If there is no time then there is no velocity and no rotation. Lots of "time" in that section!
"Interactions Velocity Limit" is a "principle of causality" fantasy that says all interactions must travel at a "c" which is not the speed of light but an undetermined value (thus a fantasy). Some nonsense to make your "c" a constant and the same in all inertial frames.
"Velocities transformation and non-conservation of cause-and-effect relations during the transition between frames of reference" is more "switch from one frame of reference to another" nonsense. Frames are switched by changing speed.
"Derivation of Lorentz transformations" is close to a lie because you use a mainstream textbook to do this, not your ATM. I will assume that you followed the mainstream textbook accurately. That gives the mainstream Lorentz transformation where c is the speed of light, not your "c".
"Physical Meaning of Lorentz Transformations and STR (Special Theory of Relativity by Einstein)" starts with your "no time" error. A "there is a violation of cause-effect relations when switching from one frame of reference to another" (in SR) error. SR uses Minkowski space rather then Euclidean space because SR is impossible in Euclidean space. A big problem is that you know this because you know about the Lorentz transformation. That has clear and obvious mixing of time and space which does not happen in Euclidean space.
"Space-Time Direct Transformation in the Hypothesis Framework." is irrelevant word salad and math.
"Principle of Locality" is word salad.
"Curved Space-Time and Gravitation" is word salad with a fantasy that gravitation (GR) is "curvature of a hypersurface", etc.
"Mass and Inertia" needs to actually say what they are!.
"Hypersurface Space-Time Metrics" is a fantasy that "a special theory of relativity has been obtained.".
"Object Motion in a Gravitational Field" is a delusion that "equation of an object free motion in the special theory of relativity" can be generalized to "an object motion equation of in a gravitational field". SR does not include gravity and strictly speaking has no acceleration. Your "curvilinear coordinates" seem not to be Euclidean space which is a requirement for your theory. Christoffel symbols are generally used in GR.
"Gravitational Field Equations" is a bit of gibberish. You start with your imaginary hypersurface. You state a general form for an action S. Some nonsense and you assume that particles do not interact with a field. A "Now we can proceed to the derivation of the gravitational field equations." fantasy when you do not derive them. What you do is take an equation that is a similar form to an equation in a mainstream textbook but with different meanings for the symbols and substitute what the textbook defined them as.

In the comments, I see one clear and big problem – it is comments not about my theory, it is comments about misconception of the theory.

And key misconception is right at start:

Originally Posted by Reality Check

"No time" is extremely wrong because time is what separates event. For example it took time separated the start of my typing rom the end of my typing. SR and GR explicitly have time as a 4th dimension. No time = no time dimension and makes it is impossible to derive anything in them.

So, you name key concept of theory as wrong right from start, without any consideration.
And, without understanding of the concept, rest of theory looks meaningless.

How to add time to system without time.
Let us consider plane (x,y) with defined on the surface function f(x,y)=x+y
Do you see any time dimension or dynamic here? I not see them
Next, how to add dynamic and time to the model with plane with defined on the plane function.

Quite easy to do. Take some line. Next, find how function on following line, after parallel shfit, would look. It is possible to find function which depend on distance between lines (it cannot be done in generic case, but lets consider case when it is possible).
If take vertical line x=2 , how values of function would look for parallel line on distance s? Equation of the line is x=2+s
For each point y, belonging to line x=2, value at lines x=2+s at same y would be: f(x,y) = x+y = (2+s) + y = (2+y) + s = f(x=2, y) + s
So, if consider how values of function evolve during parallel transfer of line at point y, the value depends on s, distance between parallel lines. It is parameter of evolution in equations. Time is also parameter of evolution in equation. So I take the parameter, distance and use it as time. As result, instead of 2d plane, I got 1-d space and time.
If take not vertical line, time can be added in same way. Just, instead of looking how function evolve at point y, necessary to look how function evolve along line, perpendicular to chosen line.

In my theory I use not distance between lines, but distance between 3-d hyperplanes and, later between 3-d hypersurfaces. No any changes happens with defined on 4-dimensional euclidean space fields. So, there is no time and there is no dynamic on fundamental level of the theory. Fundamental fields in my theory, similar to function f(x,y)=x+y, not change over time, because time is not exists. How it is compatible with quantum mechanic, let’s put aside of the discussion.
The model looks very counter intuitive, Such introduction of time looks more as mathematical trick. But what if it is correct guess? And in order to validate it, it is necessary to look for consequence of such time.
And the article explain the consequences.

So, you name key concept of theory as wrong right from start, without any consideration....

I stated your fatal error of "no time" and the easily understood reason why it is wrong. In physics and the real world (not the math of an (x,y) plane), time exists. Worse is that you say there is no time, energy, or dynamics. That does not describe this universe and is not physics. If some magical way you can get from your "no time, energy, or dynamics" to physics that has the time, energy, and dynamics of this universe then you will be doing physics.
A formal question to show the "not time" error:IF01: Give your definition of velocity that has no time, Ans, or derive the definition of velocity that we use that has time from your theory.
That will not be a story using words such as "evolve" or change" or "transfer" that have time in them as in this post. No time = nothing evolves or changes or can be transfered.

I stated your fatal error of "no time" and the easily understood reason why it is wrong. In physics and the real world (not the math of an (x,y) plane), time exists.

First, you mixing here two assertions:
1. Time exists in physics
2. Time exists in real world

Second, you assertions looks incomplete. I guess you mean “time exists and it is real phenomenon”.

First, lets look at time in physics. Time in physios is parameter of evolution in equations. Time in physics, in current mainstream physics, cannot be quantized.
All of it is not prohibit deriving time in way as I did for plane (x,y).

Second, about time exists in real world and it is real phenomena. And here we are go into philosophy. Debates about realism vs idealism go for very long time. So far, realism was not proven. It means, there is no prove that time is real phenomenon. It is possible to use argument about “time is real” as argument against any realism-based theory. However, my theory is not based on realism, it is based on subjective idealism. So, the argument cannot be used against the theory. You basically saying: “the theory is not based on realism”. And I answer: yes, it is not based on realism.
Such theory is very far from mainstream. Realism is mainstream. However, we are in ATM forum, so it is ok to go against mainstream. As long as non mainstream theory is not contradicts to observations and have, at least potential, predictive power.

So, in case if derive equations where it is possible to use some parameter as parameter of evolution, it is possible to try to use the parameter as time. If such time is consistent with SR and GR – it is strong candidate for explanation of nature of time.

Originally Posted by Reality Check

Worse is that you say there is no time, energy, or dynamics. That does not describe this universe and is not physics.

As I shown above, it is simply philosophical bias, the assertion have no any scientific basis.

Originally Posted by Reality Check

If some magical way you can get from your "no time, energy, or dynamics" to physics that has the time, energy, and dynamics of this universe then you will be doing physics.

As I see, in order to do so without fundamental time and dynamic I need to show following:
1. It is possible to add something which act as parameter of evolution to system without time and dynamic
2. It is possible to add observer to the system (most tricky part of the theory, and you not even asked question about the part)
3. It is possible to add motion and dynamic to the system. And, of course, the system should not have any time or dynamic, as f(x,y) in previous example have no time and no dynamic
4. It is possible to derive equations of SR for the time and space
5. It is possible to derive equations of GR for the time and space
And, as far as I can see, it was already done in my article. If it was done correctly and without logical errors – I am doing physics.

Originally Posted by Reality Check

A formal question to show the "not time" error:IF01: Give your definition of velocity that has no time, Ans, or derive the definition of velocity that we use that has time from your theory.
That will not be a story using words such as "evolve" or change" or "transfer" that have time in them as in this post. No time = nothing evolves or changes or can be transfered.

Easy and simple.
Note – I would use time as I derived in my theory. Example how I derive it on system without time I already shown.

Let’s look again on plane (x,y) with defined on the surface function f(x,y)=x+y
Next, take vertical line x=2. Look at parallel line at distance s: x=2+s.
As I already shown: f(s) = f(s=0) + s
How to add velocity v: y(s) = y(x=2) + v*s
y(x=2) is value of y at line x=2. y(s) is value of y at line x=2+s
It is exactly match classical equations of velocity.

Originally Posted by Reality Check

No time = nothing evolves or changes or can be transfered.

And about the assertion I already wrote it is not scientific assertion. I provided equation with velocity, which is exactly match classical equations of velocity. I see only several ways how ti can be refuted:
1. Provide prove of realism
2. Find some logical error. And find it not from point of view of realism, but with usage of scientific arguments
3. Show that the equations not fits with observations, .

This is the definition of velocity in 1 dimension x : v = dx/dt where dx is an infinitesimal change in position, dt is a infinitesimal change in time. That is not "y(x=2) is value of y at line x=2. y(s) is value of y at line x=2+s". That is simply 2 points on an (x,y) planes. There is no change in position (dx). There is no change in time (dt) because there is no time!

This is the definition of velocity in 1 dimension x : v = dx/dt where dx is an infinitesimal change in position, dt is a infinitesimal change in time. That is not "y(x=2) is value of y at line x=2. y(s) is value of y at line x=2+s". That is simply 2 points on an (x,y) planes. There is no change in position (dx). There is no change in time (dt) because there is no time!

Again, you look on it from realism point of view.
In order to understand where time here, it is necessary to understand where observer in the theory.
I postulate that observer can exists in space-time, derived in described above way.
For the example with plane (x,y), space at time s would be line x=2+s. For s=0 it is x=2
From point of view of such observer, who lives in 1d space, he would observe exactly what you described: “Velocity in 1 dimension x : v = dx/dt where dx is an infinitesimal change in position, dt is a infinitesimal change in time.”
Postulates normally are not discussed, but necessary to analyze consequences of postulates.
The postulate is point where my theory differ from realism. From point of view of external observer, there is no any changes in the system without time. However, external observers are not possible for the model. Only internal observers can exists. And for internal observers, their observations match to what we see.

For your extended example: You have an observer observing 2 unchanging points on an (x,y) plane.IF02: How can your observer who is in a space with no time measure a change in x (the dx in velocity) when all you have is 2 unchanging points, Ans?
Also note that the mathematical definition of velocity is that it is the change in x divided by the change in time in the limit as the changes go to zero. It is the slope of a tangent line touching a distance versus time curve, not 2 points. You must transform your (x,y) space to a (x,t) space to have velocity. It is that missing step that makes everything else in your idea useless.

For your extended example: You have an observer observing 2 unchanging points on an (x,y) plane.IF02: How can your observer who is in a space with no time measure a change in x (the dx in velocity) when all you have is 2 unchanging points, Ans?

In order to answer the question, it is necessary to look more close on observer.
Observer, as we know, must have some body.
Where is body of the observer in the model?
If take vertical line x=2, the body must be on the line. Only on the line. It would evolve perpendicular to the line, as value s, which I use as emergent time, increasing.
It is possible to say what states of body of observer at following point of emergent time s depends on states of the body at previous time. As we can see, the function f(x,y)=x+y can be rewritten in such way. Of course, the function is too simple to really support states for observer, but we talking more about how it can be done in principle.
I postulate that observer in such emergent time would be able to feel that it really exists.
The observer would not be able to observe anything which is not on line with his body at any given moment of his emergent time. So, here from (x,y) become (x,t)

Note that an answer is that you cannot answer the questions and so currently your theory is disconnected from the universe we live in. That is a fatal error making the rest of your theory useless. You should ask for the thread to be closed while you fix it or you might run out of time.

Reality Check, thank you for the reply and proposal to think and fix the theory.
However, the theory is well-thought.
Main problem which I see is not problems with theory, it is problem how to explain the theory. Well, improve argumentation for typical questions is one of my goals, additionally to test theory.

In my theory observer differ from observer in SR and from observer in QM. It must be intelligent being only.

About events.

Originally Posted by Reality Check

For your extended example: You have an observer observing 2 unchanging points on an (x,y) plane.

You asking about events in space without time. The question is wrong. Events happens in space-time only. And, because you asking about events on system, without time, you again and again asking about where is time is system without time.
Answer is quite obvious – system without time have no time.
System without time have no events.

However, instead of time for entire system (which is absent) it is possible to introduce internal time. The time is parameter of evolution of subsystem, defined on the space of system. As I shown for line x=2 and f(x,y)=x+y, it is possible to do and it is possible to find parameter of evolution. And it is possible to add velocity here, same as usual for (x,t)
You rejecting the idea without any consideration of it, asking for usual time in system without that usual time.

As for events. Events happens in space-time only. So, events cannot happen in (x,y). They can happen in (x,t). And the (x,t) were already introduced.

So, as it looks for me, both questions were answered. But in order to understand the answer, it is necessary to really consider proposed (x,t) spacetime, not rejecting it without consideration.

Ans, asserting the existence of a 4th spatial dimension is meaningless. Do you have any evidence that such a thing exists?

I use 4th spatial dimension as time.
There is field defined on the 4 dimensional space without time and dynamic. The field have no any quantum properties, and it is no changing over time, because time and dynamic is absent. The field can be described by some partial differential equation. Value of field at each point of space affect values at each other point in entire 4 dimensional space.

Slice space into 3d hyperplanes. The hyperplanes have some value of projection of fundamental field. Field should have such properties, what it should be possible to have some automorphism for the hyperplanes. These automorphism I use as time.
Fundamental 4 dimensional space with defined on the space field have no preferred direction. All directions have same properties.
It is possible to rotate 3d hyperplane by some angle and find new automorphism/time for rotated hyperplane. So, there is no dimension dedicated for time.

Observer exists in 3 dimensional hyperplane which is observable space for observer at some moment of its time. And, as I postulate, it is possible for intelligent being exists and feel its existence in such system without time and fundamental level and with such internal time.
I not postulate what any system without time can have intelligent observer, I postulate only possibility of it if other requirements are met.
Without observer as intelligent being, the 3d space slices with automorphism would be just mathematical abstraction. So, it is necessary to add observer as intelligent being. Observer observe such emergent space with internal time, and it bring the spacetime into being. Here comes anthropic principle as result of such model.

The described above model may looks as having some obvious weaknesses, like it may look as incompatible with SR, principle of locality etc. But it is only on first glance. If look in more details, the model is fully compatible with SR and GR and with principle of locality.

So basically you are back at regular 4-space then, but you just call it differently.

No. For one 3d hyperplane dimension for time would be one which is perpendicular to the hyperplane. If rotate the hyperplane by some angle, dimension for time would be one which is perpendicular to the rotated hyperplane. So, any dimension can be used as time and any dimension can be used as space.

No. For one 3d hyperplane dimension for time would be one which is perpendicular to the hyperplane. If rotate the hyperplane by some angle, dimension for time would be one which is perpendicular to the rotated hyperplane. So, any dimension can be used as time and any dimension can be used as space.

Please show your math.

"I'm planning to live forever. So far, that's working perfectly." Steven Wright

Well, math is in article.
What is quite funny, in order to derive SR and GR in my theory, I use math from existing mainstream textbooks. So, I can give links to that textbooks. But I guess it is not really needed.

So, I use existing math from mainstream textbooks to try to defeat mainstream. Looks strange, right?
How I do it: I derived from model of my theory following results:
1. Space and time homogeneity
2. Space isotropy
3. Presence of the maximum interactions velocity

I do it right from model of theory. If there is any logical error, it is in deriving that conclusions. But so far, I not see any.

And next, I simply copy-pasted existing math to derive SR.
The article is only foundation of more general theory. All what I want to do in scope of the article is to show what my theory is compatible with SR and GR.

Before someone who understand special relativity well enough will come and say: “it is not possible to build spacetime with Lorentz metric in 4d space, so the theory is obviously wrong” I would answer it. Yes, I fully aware that it looks impossible. It is simple mathematical exercise to prove it. However, it is not applicable to my theory. Reason: in my theory there is no requirement for event to be same in all frames of references. In my theory event may happen in one frame of reference and not happen in another frame of reference. That prove of inability to build spacetime with Lorentz metric in Euclidean space is based on assumption that all events, if happens, must happen in all frames of reference.

You've been warned several times to follow our rules and I did so very plainly in my last private message to you. If you have the math, present the math. You can make specific references to your article but you may not simply wave to the article and tell people to read it. If you continue to violate our rules, you will find yourself suspended again and your thread will be closed.

Assrtion Of Insertion

I think your theory has a big problem. The mainstream already has an excellent time dimension.

Do you have any testable predictions?

I'm not a hardnosed mainstreamer; I just like the observations, theories, predictions, and results to match.

"Mainstream isn’t a faith system. It is a verified body of work that must be taken into account if you wish to add to that body of work, or if you want to change the conclusions of that body of work." - korjik

ok, math to derive Lorentz transofrmations (it is quite badly formatted, not found how to insert latex here). Also note, most of text below was copy-pased from existing textbook.

Derivation of Lorentz transformations
To derive Lorentz transformations, one shall need:
1. Space and time homogeneity
2. Space isotropy
3. Presence of the maximum interactions velocity
All three components in this hypothesis are available. Accordingly, to show how the Lorentz transformations are derived in this hypothesis, one shall just need to choose one of several known methods for their obtaining. When deriving, I use the textbook Classical Field Theory, by L.D. Landau and E.M. Lifthitz. In this hypothesis, there are no assumptions about what the maximum velocity is. I will
assume that the maximum velocity is equal to the velocity of light. If the resulting equations coincide with the well-known Lorentz transformations, then this will mean that the maximum velocity is equal to the velocity of light.
Linearity of Transformations Due to the homogeneity of space and time and the isotropy of space and the principle of relativity, the transformations from one IFR (inertial frame of reference) to another should be linear. The linearity of the transformations can also be inferred, assuming that if two objects have the same velocities relative to one inertial frame of reference, then their velocities will be equal in any other IFR as well (it is also necessary to use weak assumptions about the differentiability and mutual uniqueness of the transformation functions). If we use only the IFR “definition": if some object has a constant velocity relative to one inertial frame of reference, then its velocity will be constant relative to any other IFR, then we can only show that the transformations between two IFRs should be linear-fractional functions of coordinates and time with the same denominator. Thus, if 𝑥′⃗⃗⃗⃗ — is the space-time vector in the system 𝑆′, and 𝐴 - is the matrix of the desired linear transformation, then 𝑥⃗=𝐴𝑥′⃗⃗⃗⃗. The transformation matrix can depend only on the relative velocity of the considered IFRs, i.e., 𝐴=𝐴(𝑣⃗).
Interval
The interval between arbitrary events is the square root of the following variable: Δ𝑠2=𝑐2Δ𝑡2−Δ𝑥2−Δ𝑦2−Δ𝑧2
where Δ𝑡=𝑡2−𝑡1, Δ𝑥=𝑥2−𝑥1, Δ𝑦=𝑦2−𝑦1, Δ𝑧=𝑧2−𝑧1 — are the time and coordinates differences of two events. If the interval between events is equal to zero in one IFR, this means that the time period Δ𝑡 - is the time (in this IFR) of the light signal travelling over distance of 𝑙 between the spatial coordinates of these points. In another IFR, it travels the distance between these points (the length of this distance - 𝑙′) for some other time period Δ𝑡′, therefore, the velocity, multiplied by Δ𝑡′ should also be equal to 𝑙′. However, the light signal velocity is the same in all IFRs; therefore, in the second IFR, the interval will also be equal to zero. Thus, directly from the velocity of light equality in all frames of reference, follows the statement:
if Δ𝑠2=𝑐2Δ𝑡2−Δ𝑥2−Δ𝑦2−Δ𝑧2=0, then in any other IFR Δ𝑠′2=𝑐2Δ𝑡′2−Δ𝑥′2−Δ𝑦′2−Δ𝑧′2=0
For infinitely near events, we have 𝑑𝑠2=𝑐2𝑑𝑡2−𝑑𝑙2 and 𝑑𝑠′2=𝑐2𝑑𝑡′2−𝑑𝑙′2. Let us assume that 𝑑𝑠′2=𝑎𝑑𝑠2. In particular, if 𝑑𝑠=0, then 𝑑𝑠′=0 as well. Due to the homogeneity and isotropy of space and time, 𝑎 cannot depend on spatio-temporal coordinates, but can only depend on the relative velocity of frames of reference. It should also not depend on the direction of relative motion due to the isotropy of space. Due to the principle of relativity, the functional relation of the dependence on the relative velocity should be universal, it is the same for all IFRs. Let us consider three frames of reference 𝑆, 𝑆1, 𝑆2, where the movement velocity vectors 𝑆1 and 𝑆2 in the system 𝑆 are equal to 𝑣1⃗⃗⃗⃗⃗ и 𝑣2⃗⃗⃗⃗⃗. Let us consider some interval in these three frames of reference:
𝑑𝑠2=𝑎(𝑣1⃗⃗⃗⃗⃗)𝑑𝑠12 , 𝑑𝑠2=𝑎(𝑣2⃗⃗⃗⃗⃗)𝑑𝑠22 , 𝑑𝑠12=𝑎(𝑣12⃗⃗⃗⃗⃗⃗⃗)𝑑𝑠22
from here it follows that 𝑎(𝑣12⃗⃗⃗⃗⃗⃗⃗)=𝑎(𝑣2⃗⃗⃗⃗⃗)/𝑎(𝑣1⃗⃗⃗⃗⃗)
However, 𝑣12⃗⃗⃗⃗⃗⃗⃗ depends not only on 𝑣1⃗⃗⃗⃗⃗ and 𝑣2⃗⃗⃗⃗⃗, but also on these vectors direction, therefore this ratio is possible only if the functional relation a(v) does not depend on v at all, that is, it is some constant. It follows from the same ratio that a=1. This means that the ratio is always satisfied: ds2=ds′2
Lorentz transformations themselves can be obtained from their linearity and the requirement of invariance of the interval.
For simplicity, we shall also consider the case of one-dimensional space. The invariance of the interval means that x2−(ct)2=x′2−(ct′)2. Let’s substitute linear transformations into this formula: x=a11x′+a12ct′ ct=a21x′+a22ct′
We obtain: x2−(ct)2=(a11x′+a12ct′)2−(a21x′+a22ct′)2=(a112−a21 2)x′2−(a222−a122)(ct′)2+2(a11a12−a21a22)x′ct′=x′2− (ct′)2
Since x′ and ct′ are arbitrary, the coefficients of the left and right sides should be identically equal. Consequently
𝑎112−𝑎212=1, 𝑎222−𝑎122=1, 𝑎11𝑎12−𝑎21𝑎22=0
It follows from the last equation that a22/a11=a12/a21. Let us mark the indicated relation as α. In addition, let us mark a11=γ, a21=b. Then a22=αγ, a12=αb. Then the first two relations can be written as follows:
γ2− b2=1, α2γ2−α2𝑏2=1,
from which it follows that, firstly, α2=1, and secondly, γ2−b2=1 , whence we can write γ2=1/(1−b2/γ2)
Finally, for convenience, while introducing the designation β=b/𝛾, we obtain:
𝐴=(𝛾±𝛾β𝛾β±&#120574=𝛾(1±ββ±1), 𝛾=±1√1−β2
moreover, the signs in the matrix are either positive or negative at the same time. The sign in the formula for γ should be chosen positive, because, at zero relative velocity of the systems, the matrix A should be unitary (the systems are identical in this case and the transformations are identical). In addition, if the coefficient in gamma had to be negative it would be impossible (the upper diagonal element would be -1, but should be 1). Therefore, we can clearly state that γ is a positive number.
As for the signs inside the matrix and, namely, the value of β, this can be established if we take the origin of coordinates of the system as S′ - vector (0,ct′) and transform it to system S, and use the velocity of movement agreement as follows: (𝑣𝑡𝑐&#119905=𝛾(1±ββ±1)(0𝑐𝑡′)=±𝛾(βct′𝑐𝑡′)
Having divided the first equation of this system by the second, we obtain β=v/c.
As for the sign, in view of the positivity of time, it follows from the second equation that the sign should be positive. Thus, we finally obtain as follows:
𝐴=𝛾(1𝑣/c𝑣/c1), 𝛾=1√1−𝑣2/c2
Thus, Lorentz transformations have been obtained within the framework of the hypothesis under consideration.

Yes, the 4th spatial dimension, according to my theory, cannot be directly observed in any way. Its existence can be verified only indirectly, based on predictions of the theory.

Originally Posted by John Mendenhall

I think your theory has a big problem. The mainstream already has an excellent time dimension.

And I use less number of phenomena than mainstream. Mainstream have time as fundamental phenomena, I not have it. So, Occam razor here looks better for me.
Also, mainstream have multiple problems which solves my theory. For example, my theory solve problem of gravitational singularity.

Originally Posted by John Mendenhall

Do you have any testable predictions?

Yes. But that I present here is only part of bigger theory.
In the article presented, I only show what my theory is compatible with SR and GR. It is foundation of my unified field theory. I would prefer to not discuss the bigger theory here, it is too complex for such discussion without understanding of its foundation.

I see a lot of variables called t with units of seconds in this 'timeless theory'.

The theory have no time as fundamental phenomena.
However, I derived emergent time as parameter of evolution and use it in equations.
Evolution etc is not means that there are some changes in fundamental 4 dimensional space with defined on the space field. Changes are not possible, because of absense of time and dynamic.

The theory have no time as fundamental phenomena.
However, I derived emergent time as parameter of evolution and use it in equations.
Evolution etc is not means that there are some changes in fundamental 4 dimensional space with defined on the space field. Changes are not possible, because of absense of time and dynamic.

Then you need to dial back your claims. For example you have not presented evidence that you have derived the Lorentz factor from your theory. What you have presented one of the standard derivations based on the established postulates of SR and claimed it as a success for your ideas.

For example you have not presented evidence that you have derived the Lorentz factor from your theory. What you have presented one of the standard derivations based on the established postulates of SR and claimed it as a success for your ideas.

My derivations of SR is not based on the established postulates of SR. Each of the postulates was derived in my theory first. Derived as result from model of the theory, not wrote it as postulates. It is easy to check.
May be results are exactly same, just some difference in interpretations? No, results are different. In mainstream, there is no way to build Minkowski spacetime on base of Euclidean space, such prove is quite simple. My theory allows to do it.
Another part that was done in my theory is derivation of equations of general relativity. And, again, results are exactly match equations of GR. However, there are difference in how they were derived.
In standard derivation in textbooks principle of equivalence is postulated. I derived it, not postulated.
In standard derivation in textbooks action is postulated. I derived it, not postulated.
There is clear and simple explanation why tensor of energy-impulse have no part with gravity.
Standard GR have problems with gravitational singularities. My theory, while have same equations of gravity, have no such problems, it is resolved.
And there are lot of such differencies.

So, my theory is not just another interpretation of existing theories.

So far my theory is not refuted.
How it can be done, in easy and simple way?

Well, let’s look on some opinion:

Originally Posted by Reality Check

The anthropic principle is not a scientific principle that can be derived. It is a philosophical consideration. We note that the universe allows for life in the current era and that we exist.

Here is assertion that anthropic principle cannot be derived.
But I derived it in my theory. So, there is clear contradiction. In case if my derivation of anthropic principle contains logical errors, in case if anthropic principle was not really derived from model of my theory, it would be refutal for the theory. Sounds easy and simple, right? Just show logical error.

Similar about principle of causality. I also derived it in scope of my theory, and if show that my derivation of the principle is wrong, it would be refutal of the theory.

Originally Posted by Reality Check

As soon as you write "four-dimensional Euclidean space" you cannot derive anything in relativity because it is mathematically impossible for relativity to exist in a Euclidean space. That is especially true for general relativity.

Another point which looks as easy and simple way to refute my theory. Hmm, where I mentioned that I know that it looks impossible? Right in introduction part of article.
Prove that this prove of impossibility applicable for my theory, and it also would be refutal for my theory

However, while it may looks as it is easy and simple to refute the theory, it is not.

So far my theory is not refuted.
How it can be done, in easy and simple way?

I have seen a number of claims, but it isn’t clear to me that you have presented a falsifiable hypothesis.

Could you state just what, specifically, your hypothesis is, some non-trivial predictions it makes that aren’t made by existing theory, and what evidence you have found that supports these predictions?

Well, let’s look on some opinion:

Here is assertion that anthropic principle cannot be derived.
But I derived it in my theory. So, there is clear contradiction.

Why should we care if you can derive some form of the anthropic principle from your arguments? How does this this lead to scientifically measurable results that can be used as a test for a hypothesis?

Much like Reality Check, I don’t see where the science is here or the point of bringing up anthropic principle arguments when discussing what is supposed to be a scientific hypothesis.

Similar about principle of causality. I also derived it in scope of my theory, and if show that my derivation of the principle is wrong, it would be refutal of the theory.

Again, I am more interested in what testable predictions you can make that support your claims.

However, while it may looks as it is easy and simple to refute the theory, it is not.

I can make a claim that there is an invisible elf in my yard, but should anyone care that it is not easy to refute? What is more important is how the claim can be supported and if there is any actual support for the claim.

"The problem with quotes on the Internet is that it is hard to verify their authenticity." — Abraham Lincoln

I say there is an invisible elf in my backyard. How do you prove that I am wrong?

My derivations of SR is not based on the established postulates of SR. Each of the postulates was derived in my theory first. Derived as result from model of the theory, not wrote it as postulates. It is easy to check.

Originally Posted by Ans

However, the light signal velocity is the same in all IFRs

Originally Posted by Ans

Due to the principle of relativity, the functional relation of the dependence on the relative velocity should be universal, it is the same for all IFRs.

Postulates one and two are in your derivation. So you claim you derived these from your theory first? Can you show that and show your derivation of the postulates you have listed? And no, I am not going to vixra to read anything. Present it here.

Originally Posted by Ans

Another part that was done in my theory is derivation of equations of general relativity. And, again, results are exactly match equations of GR. However, there are difference in how they were derived.
In standard derivation in textbooks principle of equivalence is postulated. I derived it, not postulated.

And please show this while you are at it. It is wort noting that 'action' is not really a postulate. It falls out of the more generally accepted foundations of GR - that the theory must be generally covariant, spacetime is described by a pseudo-Riemann manifold of sign -+++ and that the Einstein field equation describes its curvature. I suppose you could argue that the principle of least action is a postulate - is that what you were referring to?

I'm asking because you have made a number of big claims and when pushed just copy pasted standard derivations that could be made without your theory. You've left out the important bits - I'm pretty sure people are fairly familiar with things like deriving the Lorentz relationship from basic postulates. If they are not Wikipedia and ten minutes fixes that. So I am not sure what the value in you presenting that here is. The important bit is to back up your claims that the underlying postulates can be derived from your ideas.

Originally Posted by Ans

So, my theory is not just another interpretation of existing theories.

Based solely on what you have presented here it is not even that. It is just a claim that if we assume t is not a time dimension but something that comes from your theory, but behaves just like a time dimension, then physics works. Which I have to say is an underwhelming insight.